Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#1408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #353 |
| commutator | C42 | C2×C4×C7 |
| Frattini, cosocle | #127 | C22×C7 |
| prime radical | C7×#493 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×#127 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,17,18,13)(11,15,21,14)(12,16)(19,22,20,23);(8,10)(9,13)(11,20)(12,19)(14,22)(15,17)(16,23)(18,21)119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 540);